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Asymptotics of the potential of the electrostatic field in regions with fine-grained boundaries. (English. Russian original) Zbl 0594.35030

This paper deals with the electrostatic field \(u^{(N)}(x)\) of given charge density distribution f(x) between N disjoint conductors \(F_ i^{(N)}\), \(i=1,...,N\) whose (constant) potentials are \(C_ i^{(N)}\) with respect to a surrounding conducting closed surface \(\partial \Omega\), namely \[ \Delta u(x) = -4\pi f(x) \text{ in }\Omega -\cup^{N}_{i=1} F_ i^{(N)}\subset R^ 3, \]
\[ u=C_ i^{(N)}\quad on\quad \partial F_ i^{(N)},\quad \oint_{\partial F_ i^{(N)}}(\partial u/\partial \nu)ds=0,\quad i=1,...,N,\quad u=0 \text{ on }\partial \Omega. \] The asymptotic behavior of the solution \(u^{(N)}(x)\) is investigated when the number N of inclusions \(F_ i^{(N)}\) grows to infinity while their diameters and the distances between them tend to zero in such a way that the total volume of the inclusions remains positive. The result is that \(u^{(N)}(x)\to_{N\to \infty}u(x)\) under appropriate conditions where u(x) is the solution of the quasilinear boundary value problem \[ \sum^{3}_{i,k=1}(\partial /\partial x_ i)(a_{i,k}(x)(\partial /\partial x_ k)u(x))=-4\pi f(x)b(x) \text{ in } \Omega,\quad u=0 \text{ on } \partial \Omega, \] for some continuous functions \(a_{i,j}(x)\) (positive definite) and b(x).
Reviewer: G.Philippin

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
31A35 Connections of harmonic functions with differential equations in two dimensions
35B40 Asymptotic behavior of solutions to PDEs
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